Growing a shape with changing color
Well, you didn't tell which render engine you need, but I have a "tricky" solution which I here explain using Bender Render, but it's obviously doable even with Cycles Render. But, I used a trick, and I don't know if you can use it, too.

here's the result (colors have banding just for the gif format...):



how I did it?

First I created a plane, scaled it and then subdivided it a lot, 
and assigned it a shadeless material made by a blend texture with a color ramp, note the settings below:



then,I created a half circle bezier with its origin in the same place as the previous plane



then I've set a curve modifier for the plane, to the bezier (note the X scale, later on this)



then, X scaled it until it covered the bezier, and set frame 1



then, duplicated it to plane.001, scaled it a bit on its Y (to widen its curve, so to cover the underlying rainbow curve), and then I keyframed the same X scale at frame 1



then, I set a dark shadeless material which is the same as the world background, and set a key frame for X scale = 0 on last frame (50 in my example), so that during the animation it "uncovers" the below rainbow arc...



here is last frame



so, when you start the animation you see this progressing







and the result of the animation is what you see in the first animated gif...

here is my test file 


Of course your need could be slightly different, but you could use this as a starting point...

edit: some time ago I answered another (somewhat related) question,and there I did both a Blender Render and Cycles Render colorramp... see How to make / generate the trailing effect?